Modified Legendre Operational Matrix of Differentiation for Solving Strongly Nonlinear Dynamical Systems

  • A. K. Alomari
  • Muhammed SyamEmail author
  • Mohammad F. Al-Jamal
  • A. Sami Bataineh
  • N. R. Anakira
  • A. F. Jameel
Original Paper


Complex vibration phenomena appear so frequently in many engineering and physical experiments, and they are well modeled using nonlinear differential equations. However, contrary to the linear models, nonlinear models are difficult to analyze analytically or numerically and particularly for long-time spans. In this paper, we propose a novel method to provide approximate analytic solutions of an important class of nonlinear differential equations that describe the underdamped, overdamped, and oscillatory motions of massspring systems subjected to external excitations. The method is based on a novel modification of the Legendre operator matrix of differentiation technique which results in solutions that are accurate not only for short-time spans but also for long-time spans as well. We provide error analysis and present several examples to demonstrate the efficiency of the proposed method.


Mass-spring systems Nonlinear differential equations Chaotic systems Legendre operator matrix of differentiation 

Mathematics Subject Classification

34K28 65P20 32A05 



The authors would like to thank the reviewers for the valuable comments.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  • A. K. Alomari
    • 1
  • Muhammed Syam
    • 2
    Email author
  • Mohammad F. Al-Jamal
    • 1
  • A. Sami Bataineh
    • 3
  • N. R. Anakira
    • 4
  • A. F. Jameel
    • 5
  1. 1.Department of Mathematics, Faculty of ScienceYarmouk UniversityIrbidJordan
  2. 2.Department of Mathematical SciencesUAE UniversityAbu DhabiUnited Arab Emirates
  3. 3.Department of Mathematics, Faculty of ScienceAl-Balqa Applied UniversityAs-SaltJordan
  4. 4.Department of Mathematics, Faculty of Science and TechnologyIrbid National UniversityIrbidJordan
  5. 5.School of Quantitative SciencesUniversiti Utara Malaysia (UUM)SintokMalaysia

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